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The given expression is:

\[
\frac{-\frac{1}{2}x(1 - x)^{-\frac{1}{2}} + (1 - x)^{\frac{1}{2}}}{x^2}
\]

Let's break this down step by step.

### Step 1: Simplifying the numerator
The numerator is:

\[
-\frac{1}{2}x(1 - x)^{-\frac{1}{2}} + (1 - x)^{\frac{1}{2}}
\]

There are two terms here:

- The first term is \( -\frac{1}{2}x(1 - x)^{-\frac{1}{2}} \)
- The second term is \( (1 - x)^{\frac{1}{2}} \)

### Step 2: Dividing by \( x^2 \)
Now we divide the entire numerator by \( x^2 \):

\[
\frac{-\frac{1}{2}x(1 - x)^{-\frac{1}{2}} + (1 - x)^{\frac{1}{2}}}{x^2}
\]

We can break this up as:

\[
\frac{-\frac{1}{2}x(1 - x)^{-\frac{1}{2}}}{x^2} + \frac{(1 - x)^{\frac{1}{2}}}{x^2}
\]

### Step 3: Simplifying each term
#### First term:
\[
\frac{-\frac{1}{2}x(1 - x)^{-\frac{1}{2}}}{x^2} = -\frac{1}{2} \cdot \frac{(1 - x)^{-\frac{1}{2}}}{x}
\]

#### Second term:
\[
\frac{(1 - x)^{\frac{1}{2}}}{x^2}
\]

Now the simplified expression is:

\[
-\frac{1}{2} \cdot \frac{(1 - x)^{-\frac{1}{2}}}{x} + \frac{(1 - x)^{\frac{1}{2}}}{x^2}
\]

Would you like to proceed with further simplification, or need an explanation of any step?

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### Follow-up questions:
1. How do negative exponents affect the simplification of terms?
2. What role does the factor \( \frac{1}{2} \) play in the first term?
3. How can \( (1 - x)^{-\frac{1}{2}} \) be rewritten for easier manipulation?
4. What is the difference between \( x^{-1} \) and \( x^{-2} \) in terms of division?
5. Would factoring common terms from the expression help in simplifying further?

### Tip:
When simplifying expressions with fractional exponents, remember to handle the base terms first before dividing by any additional variables.

-\frac{1}{2} x (1 - x)^{-\frac{1}{2}} + (1 - x)^{\frac{1}{2}} / x^2

This tutorial simplifies the expression involving fractional exponents and division. The problem uses the exponent rules and algebraic techniques to break down each term, making it ideal for high school students learning advanced algebra concepts.

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